Language Model Scaling Laws

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15 papers in the last four weeks, up 200% on the four weeks before. 0.1% of all new papers.

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Latest papers 95

Oct 7, 2026cs.LG

Fault-tolerant foundation models

Emerging computer hardware often trades reliability for energy efficiency; here we show that large-language models (LLMs) can be trained to tolerate this unreliability, and that rather than degrading, their error resilience actually increases as they grow. Modified neural scaling laws inferred from 40,000 GPU-hours of training runs on simulated faulty digital hardware quantify this trend and suggest that models learn to compute within "good" error-correcting codes, whose relative overhead remains finite no matter how large the model gets. This finding leads us to conjecture that appropriately trained LLMs may be formally fault-tolerant; if true, running AI inference on low energy, faulty hardware may be a path to substantial energy savings over the status quo.

Trevor McCourt, Ila R. Fiete, Isaac L. Chuang

Department of Electrical Engineering and Computer Science, MIT · Department of Brain and Cognitive Sciences, McGovern Institute, MIT · Department of Physics, MIT

LLM Training Language Model Scaling Laws Energy-Efficient ML

Oct 6, 2026cs.AI

Toward Alignment Scaling Laws: A Framework and First Preregistered Measurements

Whether alignment gets easier or harder as models grow is often argued from isolated findings, as if alignment were one property. We treat it as a family of measurable scaling relations: for each risk category r, the alignment burden needed to hold a fixed safety target is modeled as B_r(N)=a_rN^alpha_r, with N a capability proxy; against a budget proportional to N, scaling helps if alpha_r<1, keeps pace if alpha_r~1, and accumulates alignment debt if alpha_r>1. We give three operationalizations of burden and distinguish observed, audited and true alignment. A toy model, in which corrections consume capability headroom, makes the consequences explicit. We prove that the largest exponent among corrected risks, not an average, sets the long-run regime; that above 1 any policy holding headroom above a floor must grow super-exponentially; that, for burdens that are positive mixtures of power laws, fits on small models underestimate large-scale exponents; and that an audit that uncovers hidden failures without false positives never underestimates true alignment. We propose a pre-registrable protocol and apply reduced versions of it twice. A preregistered reanalysis of public adversarial-training data for Pythia classifiers finds that the compute needed to bring attack success under 10% grows as N^0.60. A preregistered pilot on Qwen2.5 0.5B-72B finds exponents of -0.05 for truthfulness and 0.48 for stated dispositions (both scaling helps under its reduced rule, though local slopes approach 1 at the top; replicated on Qwen3 0.6B-14B), while sycophancy (0.89, or 0.83 with two seeds added at 72B) and a planted backdoor are undetermined: the backdoor is removed quickly when its trigger is known but survives blind safety training at four of five sizes. We release four browser games that play these laws (www.aisafety.fun). We make no claim about which regime holds for current frontier models.

Jeremy Canale

LLM Alignment Language Model Safety Evaluation Language Model Scaling Laws

Oct 5, 2026cs.LG

Scaling Down the Scaling Laws: Parameter Efficiency and Compute-Optimal Training in Resource-Constrained Large Language Models

Large language models (LLMs) have achieved substantial performance gains through increases in model size, training data, and computational resources. However, traditional scaling approaches produce diminishing returns, rising financial and environmental costs, and barriers to participation for researchers operating outside large industrial laboratories. This review examines the evolution of LLM scaling theory from empirical scaling laws to compute-optimal training, with particular emphasis on parameter efficiency, token utilization, data efficiency, and resource-constrained environments. Foundational work on scaling laws is synthesized alongside later research on compute-optimal training, data pruning, efficient architectures, quantization, low-rank adaptation, and edge-oriented optimization. The literature indicates a shift from scale maximization toward more deliberate allocation of parameters, tokens, compute, and hardware resources. At the same time, important empirical, theoretical, and methodological gaps remain regarding whether scaling principles established on enterprise-grade infrastructure generalize to smaller models and constrained computing environments. This review organizes these developments into a unified framework for resource-efficient LLM training and argues that future progress should evaluate efficiency not solely through model performance, but through the relationship among performance, parameter count, computational cost, token allocation, and hardware constraints.

Joe Dwyer

ECPI University

Language Model Scaling Laws Efficient Language Model Training

Oct 4, 2026cs.CL

What Is a Repeated Token Worth? The Scaling Geometry of Multi-Epoch Pretraining

As pretraining increasingly repeats data, every run faces three questions: how many epochs to take, how that number should change with model size, and whether anything besides the epoch count matters. We answer them by pricing a repeated token against two references: one epoch on the same data, which gives its value, and fresh data at equal compute, which gives its cost. Against fresh data, the cost of repetition follows a single variable, the number of extra epochs divided by the unique tokens per parameter. Against the same data, a second epoch is worth nearly as much as a fresh one, and repeated tokens fall to half the value of fresh ones after a critical epoch count that grows with the training budget per parameter but hardly with model size. With unique data fixed, the predicted compute-optimal run grows model size and epochs together until loss stops improving, near the critical epoch count. The same variable accounts for the direction of size trends that appear to conflict: larger models tolerate fewer epochs when the corpus is fixed, from about 15 at 127M to 4 at 2B parameters, but not when unique data grow with the model. Counts alone do not determine loss: at identical counts, replaying shards consecutively raises loss by up to 0.46~bits per byte, concentrating repeats on fewer samples also raises it, lower-entropy sources degrade faster with repetition, and re-tokenizing repeats helps only under heavy repetition. These results offer an empirical guide to pretraining when unique data, rather than compute, are the binding constraint.

Yekun Chai, Haoyi Xiong

FloatAI · ETH Zurich · Independent Researcher

Language Model Pretraining Language Model Scaling Laws Data Repetition in Language Model Training

Oct 4, 2026cs.CL

Selecting Repetition Counts Across Model Scales in Data-Constrained Pretraining

The repetition count that works best for a small language model may not remain best at a larger scale. We study this effect in pretraining with a finite target corpus mixed with generic data at a fixed target fraction. On Wikipedia-derived data and Proof-Pile-2, the ranking of measured repetition counts changes with model size, and a 520M Proof-Pile-2 experiment confirms that reducing repetition from sixteen to eight improves loss while using fewer training tokens. We use loss curves from several smaller models to retain a short list of promising repetition counts for evaluation at a larger scale. On PubMed and Caselaw, candidate sets fixed before target-model training retain the lowest-loss measured count on the original evaluation grids at both 200M and 520M. This supports candidate retention as a practical alternative to exact point prediction. We also relate the pruning regression to an empirical scaling model with two opposing repetition-dependent loss terms. A first-order expansion in log model size yields the linear form used by the selection rule, providing a scaling-based interpretation of the candidate-selection procedure.

Ziyue WANG, T. Kanamori

Institute of Science Tokyo

Language Model Pretraining Language Model Scaling Laws Efficient Language Model Training Data Repetition in Language Model Training

Oct 4, 2026cs.CL

A Systematic Analysis of the Predictive Power of LM Surprisal in Reading Chinese

This study analyzes the predictive power of LM-derived, token-level surprisal on Mandarin Chinese reading times. We first propose the Shortest Matching Sequence (SMS), an alignment scheme that maps between the word segmentation assumed by eye-tracking corpora and the LMs' subword tokenization, as the two tokenizations often disagree in the context of Mandarin Chinese. Then, using a suite of Chinese-Pythia models (14M-1.4B) trained on scratch with 30B tokens, we examine how well surprisal predicts first fixation duration, gaze duration, and total reading time in three paragraph-level eye-tracking corpora of Mandarin Chinese (GECO-CN, HKP, and MECO). Contrary to previous null findings, our results show that surprisal is predictive of Chinese reading times. However, whether predictive power scales with model size and the amount of training is corpus-specific: bigger models predict better in GECO-CN, whereas inverse scaling emerges in HKP and, at the largest sizes, in MECO. Subsequently, we tested one possible explanation for the inverse scaling in HKP and found that checkpoints whose surprisal remains closer to

n

-gram statistics are better predictors of reading. All in all, the predictive power of surprisal on Chinese reading time measurements is corpus-specific, which cautions against drawing scaling conclusions from a single corpus.

Hongao Zhu, Muxiaoqiao Xu, Yikang Liu +4

Department of Linguistics, UC San Diego · School of Foreign Languages, Shanghai Jiao Tong University · School of Computer Science, Shanghai Jiao Tong University +3

Language Model Scaling Laws Language Modeling Computational Psycholinguistics

Oct 1, 2026cs.CV

Not All Error Yields to Scale: Where Scaling Stops in Vision-Language Inference

Vision-language models (VLMs) face a fixed-budget trade-off between processing more visual information for fine-grained perception and using a larger language backbone for complex reasoning. Existing studies do not tell us which combination of backbone size and input resolution to deploy, especially in high-resolution deployments. To address this gap, we propose the Separable Law that describes how VLM performance changes with language backbone size and visual token count. We fit the law to measurements from 26 InternVL and QwenVL models, with language backbone sizes from 1B to 72B, on four high-resolution benchmarks with image sizes from 224 pixels to 8K. We find that the questions responding to scaling can be predicted from the skill they require, while a substantial fraction never responds at all. We also find that the two model families gain similarly from a larger backbone, while their gains from more visual tokens differ sharply. Combined with a cost law, the Separable Law gives a closed-form rule for allocating compute between backbone size and visual tokens. When deployment is limited to available configurations, the law identifies model and image sizes that perform close to the best feasible choice under the same budget. We hope our work offers a principled way to decide how much a model should be allowed to see at high resolution, given what it must reason about.

Xinye Zhao, Yunkai Dang, Yunchen Wu +1

School of Intelligence Science and Technology, Nanjing University

Vision-Language Models Efficient VLM Inference Language Model Scaling Laws

Sep 30, 2026cs.LG

Scaling Laws for Looped Mixture of Experts

Looped transformers and Mixture-of-Experts (MoE) offer complementary routes to efficient scaling: recurrence increases computational depth at fixed parameters, while MoE sparsity expands total capacity at fixed active compute. Yet existing scaling laws model recurrence or sparsity in isolation. In this work, we introduce Loop Scaling Laws, the first scaling law to jointly model recurrence and sparsity alongside model size and data. At its core is a bounded, sparsity-conditional recurrence mapping that characterizes the effective-parameter gain from looping and how sparsity raises this gain. The laws predict the held-out loss of looped models more accurately than prior alternatives, and recover the standard dense and MoE scaling laws as special cases. Beyond prediction, the fitted laws provide a principled foundation for designing looped MoE models under compute and memory constraints. Downstream evaluations further demonstrate the complementary benefits of the two axes: sparsity delivers ~3x active-parameter efficiency, recurrence yields ~2x total-parameter efficiency on reasoning, and joint scaling further advances the performance frontier. As a practical extension, we show these gains hold at trillion-token scale: at matched training compute, a looped MoE with law-derived recurrence matches a ~2x larger non-looped MoE on the reasoning benchmarks, while enabling test-time scaling through recurrence.

Yanbei Chen, Anirudh Goyal, Raghuraman Krishnamoorthi

Meta AI

Mixture-of-Experts Language Models Language Model Scaling Laws Inference-Time Scaling Recurrent Transformers AI Scaling Laws

Sep 30, 2026cs.CL

How Much Is an AI Token Worth? Scaling Laws for Wild AI-Generated Web Text

Web text makes up the majority of pretraining data and is increasingly AI-generated. After applying FineWeb quality filtering, we find that 27.5% of tokens from June 2026 web data are labeled as AI-generated by Pangram, rising to 31.1% by August. Unlike synthetic data or model-collapse setups, this wild AI text comes from many models, is written for human readers, and arrives unlabeled in pretraining corpora. How does AI text in the wild affect language model pretraining? To answer this question, we pretrain 800 language models, varying the ratio of added AI tokens to human tokens, and fit scaling laws to held-out losses on both human and AI-generated text. For data-starved models, adding AI tokens to pretraining data initially lowers loss on human text, but the benefit saturates as more are added and quickly reverses into harm. For models trained on high budgets of human text, AI tokens raise loss almost immediately, while the same number of fresh human tokens keeps lowering it. Scaling laws such as Hoffman et al. (2022) fail to predict this behavior. We propose a new scaling law with separate benefit and harm terms that allows the value of an AI token to change sign while also reducing to Chinchilla in the absence of AI text. When fit on smaller models, our scaling law predicts the effect of AI text on held-out human-text loss for models up to 3.6x larger with 41% lower error than the best existing law over all AI ratios. We recommend filtering AI text when the target is human text, repeating human text before expanding the training dataset with AI-generated web text, and reporting validation loss on human and AI text separately AI text remains valuable when the target is AI text. We release WildAI, an 83B-token corpus with AI, topic, and format labels, all 800 models and code at https://github.com/pangramlabs/WildAI.

Jenna Russell, Ben Glickenhaus, Katherine Thai +4

University of Maryland · Pangram Labs

Language Model Pretraining Synthetic Data Pretraining Language Model Scaling Laws

Sep 30, 2026cs.LG

From Spectra to Joint Schedules in LLM Pre-training: 3+3(+2) Scaling-Law Regimes

Power-law learning curves are often treated as fixed properties of a model and its data, although learning-rate and batch-size schedules can change the observed loss. We study this dependence in noisy online SGD with linear random features. Conditional on the representation, an exact Volterra equation separates two response components: a forcing term that propagates unresolved target error and a memory kernel that propagates stochastic-error injections. We prove that either component follows a power law if and only if its cumulative weighted spectral mass has the corresponding low-spectrum scaling; individual eigenvalues and target coefficients need not obey coordinatewise power laws. Under a joint schedule, intrinsic time

T_t=\sum_{s<t}η_s

controls optimization progress, while

r_t=B_t/η_t

controls noise injection. Their interaction yields sharp conditions under which a schedule preserves, changes, or destroys the clean power law, together with a memory ceiling on noise reduction. The power-law random-feature model realizes this mechanism in

3+3(+2)

propagation regimes with phase-dependent compute rates. Controlled nanoGPT experiments show that (1) learning-rate and batch-size schedules with matched

B/η

paths are nearly equivalent in intrinsic time, (2) a forcing-memory surrogate accurately predicts loss across schedules, and (3) its fitted exponents across real-world datasets identify the regime of LLMs in

3+3(+2)

map.

Yichen Wang, Fanghui Liu, Yudong Chen

Department of Computer Sciences, University of Wisconsin–Madison, USA. · School of Mathematical Sciences, Institute of Natural Sciences and MOE-LSC, Shanghai Jiao Tong University, China.

Language Model Pretraining Deep Learning Optimization Learning Rate Scheduling Language Model Scaling Laws Stochastic Gradient Descent

Sep 28, 2026cs.LG

ScAn-Bench: Evaluating Scaling Analysis Methodology

Recent progress in machine learning is driven by large-scale foundation models, where scaling laws and finding optimal scaling prescriptions for architecture, data, and hyperparameters are key in advancing the state-of-the-art. Therefore, it is surprising that no systematic study evaluates the methodology to obtain scaling laws and prescriptions across different model types. To shed light on this crucial blind spot and facilitate future research, we introduce the surrogate benchmarks ScAn-Bench-LLM and ScAn-Bench-VLM based on 4524 and 8024 checkpoints of language and vision-language model pipelines. On our benchmarks, we perform the first systematic evaluation of both data acquisition and extrapolation methodology for scaling analysis across different data modalities.

Artin Sermaxhaj, Nastaran Alipour, Donat Sinani +4

University of Freiburg · Zuse School ELIZA · Juelich Supercomputing Center (JSC), Research Center Juelich (FZJ)

VLM Evaluation LLM Evaluation Language Model Scaling Laws Scaling Laws

Sep 28, 2026cs.LG

X-MoD: Practical Scaling Laws for Sparse-Depth Routing Beyond Mixture-of-Depths

Mixture-of-Depths (MoD) enables conditional computation across Transformer depth by routing only a subset of tokens through selected layers, but its original one-sparse--one-dense alternation tightly couples total capacity to active capacity and limits sparse-depth scaling. We introduce X-MoD, a scalable sparse-depth architecture that decouples token sparsity from anchor stride, allowing total parameter count to grow while keeping active-equivalent capacity nearly fixed. To make deep sparse routing trainable, X-MoD combines dense anchors with variance-scaled layer-wise gating and depth-wise token balancing. To make this regime analyzable and usable, we formulate sparse-depth routing as a conditional architecture-design problem: given compute, context length, and active-equivalent backbone size, how should the routing configuration be chosen? We develop a practical scaling-law framework by fitting X-MoD relative to FLOP-matched dense baselines, yielding an interpretable law that decomposes performance into sparse-capacity gain, sparse-context correction, and anchor-stride interaction. The law predicts validation loss across routing configurations and reveals how context length, model scale, and anchor stride shape sparse-depth performance. We validate the architecture and law through pretraining sweeps, held-out scaling-law prediction, ablations, downstream evaluations, and comparisons with Dense, MoD, and representative MoE baselines.

Bowen Dong, Yilong Fan, Tengyu Pan +6

Tsinghua University · Tianjin University

Language Model Scaling Laws Mixture-of-Experts Models Conditional Computation

Sep 28, 2026cs.LG

Training and Inference Dynamics of PLDR-LLMs: Row-Map Collapse, Renormalization, and Predictive Reduction

This monograph develops a unified account of training and inference in Power Law Decoder Representation language models (PLDR-LLMs). Exact finite work identities decompose changes in the absolute energy of the row-centered learned map into parameter contributions, signed interactions, and numerical observation defects. Positive affine blocking retains restarts at the row-constant face, while the augmented AdamW state supplies the complete dynamical description. Predictive renormalization acts on the complete conditional training law for a single pass over distinct corpus target blocks, retaining optimizer memory, remaining data, schedule, and numerical policy. Autonomous reductions require closure; approximate reductions carry successor and emission errors. Finite-population covariance, matched physical clocks, matrix fluxes, and signed temporal energy connect row dynamics to model-wide observations. Absolute row collapse, relative row concentration, operator stabilization, and predictive accuracy are distinguished. Experiments reveal observer and optimizer dependence, reject the tested autonomous row-state candidates, and support finite conditional prediction and state-specific operator reduction. Independent single-pass families exhibit moving finite fluctuation regions without establishing a thermodynamic critical class. Conditional symmetry, head limits, covariance flows, and readout error budgets specify assumptions needed to transfer scaling laws to inference. The theory separates exact identities, conditional dynamical claims, and finite empirical findings, with proofs, selected formal checks, and compact numerical evidence.

Burc Gokden

Fromthesky Research Labs LLC, Oregon, USA

LLM Inference Language Model Scaling Laws Representation Geometry in Language Models LLM Training

Sep 23, 2026cs.LG

Does Step Law Transfer to Small-Scale Language Models? An Empirical Recalibration Below 59M Parameters

Step Law gives power-law formulas for the optimal peak learning rate eta* and batch size B* when pre-training language models. It was calibrated on models between 59M and 1B parameters; the small-model regime N < 59M was never tested empirically by its authors. This regime matters for single-GPU training, interpretability research, educational experiments, and settings where larger models are infeasible on memory or cost grounds. We test whether Step Law transfers to small language models. We consider three outcomes: H1, the original coefficients work directly; H2, the power-law form holds but with different coefficients; and H3, a power law does not describe the optima in this regime. All experiments use a single nanoGPT/TinyStories pipeline with a 2048-token BPE vocabulary, AdamW, and a warmup-cosine schedule. The optimum for each (N, D) cell is extracted from the loss surface L(eta, B) via a local quadratic approximation in log-log coordinates over the smoothed training loss. The final dataset contains 29 unique (N, D) cells and 935 analysis-ready runs. The main refit uses 25 cells (815 runs) in the working range 4 <= D/N <= 600. On the pooled data we accept H2: the functional form is preserved, but the coefficients differ from the original. We obtain eta*(N, D) = 0.0985 N^(-0.508) D^(0.238) (R^2 = 0.834) and B*(D) = 3.6 x 10^(-4) D^(0.931) (R^2 = 0.950). Step Law's structural claim that B* is independent of N is reproduced (p = 0.87), but the growth of B* with D is nearly twice as steep as in the original work. Direct transfer of Step Law systematically overestimates the optimal learning rate: the median ratio eta_SL / eta* is approximately 4.0x, with a range of 2.4x to 6.6x.

Egor Romanyukov, Timofey Novikov, Timur Shokarov +3

Faculty of Computer Science HSE University

Language Model Scaling Laws Hyperparameter Transfer Small Language Models

Sep 22, 2026cs.LG

Scaling of Capability and Efficiency at Inference Time in Large Reasoning Models

Capability and efficiency are two key dimensions of reasoning in large language models (LLMs). Capability refers to the ability to solve a given problem correctly, whereas efficiency refers to the ability to do so with limited resources. When LLMs use Chain-of-Thought (CoT) reasoning to solve problems of controlled hardness, both the number of problems solved correctly and the number of tokens required to reach a correct answer depend on problem hardness and model size. However, how these factors jointly shape capability and efficiency remains poorly understood. Here, we use hierarchical Bayesian models to evaluate the capability and efficiency of LLMs from the DeepSeek-R1-Distill model family across four classes of arithmetic and algorithmic reasoning problems. At a fixed model size, the probability of correctly solving an instance decays approximately exponentially with instance size, our proxy for problem hardness. The decay scale grows sublinearly with model size, indicating that larger models are more capable, but that capability gains diminish with scale. Output length grows as a power law with instance size, which serves as a proxy for difficulty. However, the parameters of this power law do not vary systematically with model size, suggesting that larger models do not become more efficient. Together, these findings reveal potential limitations of naive scaling as a strategy for developing more capable AI systems: capability improves with diminishing returns, while efficiency shows little to no improvement.

Moritz Laber, Zohair Shafi, Germans Savcisens +6

Northeastern University, Boston, Massachusetts, USA · Complexity Science Hub Vienna, Vienna, Austria · Santa Fe Institute, Santa Fe, New Mexico, USA

LLM Inference Efficiency Language Model Scaling Laws Inference-Time Scaling Algorithmic Reasoning Large Reasoning Models

Sep 16, 2026cs.LG

How Model Growth, Recursion, and Boundary Operators Influence Scaling Exponents

Scaling laws predict how loss decreases with increases in computation. We show, contrary to conventional wisdom, that architectural interventions can modify scaling exponents in pre-training, leading to power-law improvements in performance as computation increases. As an anchoring point, we consider the architectural formulation of looped transformers. Although not typically used in this way, looping, also known as recursive depth, provides a mechanism for model growth, by increasing the number of loops during training. Model growth, with and without shared weights, provides the biggest changes to the scaling exponents. In particular, a 7.4B model growth architecture matches GPT-3 13B on CORE with roughly

20\times

less compute, and has compute efficiency gains that increase with scale. Moreover, simply using a boundary operator in a vanilla transformer, which normalizes and injects an earlier block, also provides an exponent increase, although to a lesser extent. In the data-constrained, multi-epoch setting, standard looping has a useful regularizing effect, where we find it is compute-optimal to increase the number of loops with scale. These results can be understood through the lens of computational depth: for a given computational budget, we wish to increase the usable depth of the transformer, which can lead to efficiency gains that increase with scale.

Zixi Chen, Akshay Vegesna, Samip Dahal +1

New York University · Q Labs

Language Model Scaling Laws Recurrent Transformers

Sep 14, 2026cs.CL

Breaking the Token Ceiling: Distilling Smaller, Stronger Byte Models

Small models are made more capable through distillation from a larger one that shares their tokenization scheme. However, do distilled byte and token models behave similarly in terms of scaling trends as compute and data increases? To enable this comparison, we introduce two variants to efficiently convert token logits to Byte Logits: 1) approximate: Marginalize-It, and 2) exact: End-Of-Token. We then present the first large scale study of overtraining decoder-only dense transformer models varying two dimensions simultaneously: the tokenization scheme (Tokens, Bytes, Bytes w/ eot) and the training objective (Distillation vs. Cross-Entropy), sweeping layer-parameter-matched models with roughly 1 billion parameters up to 1 trillion bytes of data. Across eight benchmarks spanning three categories: Multiple Choice QA, Language Generation, and Machine Translation, we find that Token-1B models outperform byte models (End-Of-Token-1B and Bytes-1B) in the low-FLOP regime but eventually plateau; byte models start worse yet surpass Token-1B models with more compute, reaching a higher downstream task performance ceiling. Extrapolating the average top-1 error vs. validation BPB scaling laws predicts that, asymptotically, distilled End-Of-Token-1B outperforms distilled Token-1B by up to 4%. They are also far more data efficient, matching the performance of distilled Token-1B using only one-sixth of the training data. Moreover, by operating over a small vocabulary of 256 bytes instead of on the order of 100K tokens, they circumvent the need for top-k truncation during logit dumping, while also reducing logit storage costs to roughly one-fifth. Finally, our downstream performance scaling laws predict that our distilled End-Of-Token-1B models asymptotically surpass the Llama 3.2-1B, Gemma-3-1B-pt, and Gemma 2B models on averaged downstream tasks by up to 6.5%, 8.1%, and 2.1%, respectively.

Kalyani Marathe, Artidoro Pagnoni, Tomasz Limisiewicz +4

University of Washington, Seattle · Work done at Meta FAIR · Meta FAIR

Decoder-Only Language Models Language Model Scaling Laws Byte-Level Language Model Language Model Distillation

Sep 8, 2026cs.LG

Hyperparameter Scaling Laws Across MoE Sparsity

Mixture-of-Experts (MoE) models expand model capacity without a proportional increase in training compute, but increasing sparsity makes reliable hyperparameter transfer challenging. In this work, we show that conventional hyperparameter scaling laws are insufficient for ultra-sparse MoEs: the optimal learning rate and batch size vary with activation ratio, and these shifts cannot be explained by either total or activated parameter count alone. To characterize this dependence, we conduct 1,800 pre-training runs spanning six activated-parameter scales and models with up to 6B total non-embedding parameters, processing approximately 20 trillion tokens at a cost of 200,000 equivalent H800 GPU-hours. Our results reconcile conflicting findings in prior work by revealing two scaling regimes. At fixed sparsity, the optimal batch size follows a power-law relationship with training tokens

D

, whereas the optimal learning rate scales with training compute

C

and remains robust to the allocation between model size and data. Across sparsity levels, the activation ratio

A

enters both relationships as an additional multiplicative power-law factor. These observations lead to unified hyperparameter scaling laws that transfer across MoE sparsity levels. Large-scale evaluation shows that the scaling form outperforms alternative functional forms. On a held-out ultra-sparse MoE with 12B total parameters and only 1/64 of its experts activated, the predicted hyperparameters remain close to the observed optima, supporting joint extrapolation across model scale and sparsity. Further experiments demonstrate transfer across expert granularities and isolate the effect of activation ratio from that of total expert count.

Changxin Tian, Kunlong Chen, Jia Liu +3

Ling Team, Ant Group

Language Model Scaling Laws Sparse Mixture-of-Experts Hyperparameter Transfer Scaling Laws Mixture-of-Experts Models

Sep 1, 2026cs.LG

Efficiently Estimating Optimal Hyperparameter Scaling Laws through Power-Law Entropy Search

Optimal hyperparameter scaling laws describe how the best hyperparameters for large language model (LLM) training change with model and data scale, enabling practitioners to predict optimal configurations at production scales without expensive large-scale tuning. However, estimating these scaling laws conventionally requires exhaustive grid searches over thousands of training runs, consuming enormous computational resources. We introduce Power-Law Entropy Search (PLES), a computational cost-aware acquisition function built on multi-fidelity Bayesian optimization that efficiently estimates optimal hyperparameter scaling laws through adaptive experimentation. A key innovation in PLES is that it searches for candidates that reduce the overall uncertainty of a scaling law estimate, instead of optimizing a single objective function. At each iteration, PLES selects the candidate configuration that maximally reduces the uncertainty of the scaling law estimates per unit computational cost, naturally favoring informative small-scale experiments. We evaluate PLES on synthetic benchmarks, surrogate models fitted to real LLM training data, and actual LLM pre-training runs. Across all settings, PLES converges to accurate optimal hyperparameter scaling laws using less than one-tenth of the computational budget required by conventional grid search and other baselines.

Zhiliang Chen, Sebastian Ament, David Eriksson +4

Meta · National University of Singapore · Atomic Machines; work done at Meta

Bayesian Optimization Language Model Scaling Laws LLM Training Hyperparameter Optimization

Aug 28, 2026cs.LG

Deriving Scaling Laws for OpenEuroLLM Models: Learning Rate, Batch Size and Loss

We study the scaling behavior of learning rate and batch size in pretraining dense large language models on English-prevalent corpora. Beyond scaling jointly optimal learning rates and batch sizes, we investigate their marginal evolution with model capacity and data scale and develop a model that captures these relationships. As we employ a Warmup-Stable-Decay learning rate schedule, we further investigate the gains from learning rate annealing over a broad range of hyperparameters settings, models and data budgets, and whether the optimal learning rate and batch size transfer between the stable and decay phases. Finally, we characterize the dependence of loss on model capacity and dataset size, evaluating recently proposed scaling forms that explicitly model their interaction. We find these approaches particularly effective at capturing both undertraining and overtraining regimes across our experiments. This study establishes a first baseline and scaling procedure for the development of future OpenEuroLLM models. We open-source the complete collection of pretraining runs used in this study.

Niccolò Ajroldi, Diana Alexandra Onutu, Haider Al-Tahan +4

ELLIS Institute Tübingen · OpenEuroLLM · Eindhoven University of Technology +6

Language Model Pretraining Learning Rate Scheduling Language Model Scaling Laws Scaling Laws

Aug 27, 2026cs.CL

Puro-2B: Poor Lab's Qwen2-1.5B Trained on RTX 5090 within $5090

Language model pretraining has become almost synonymous with prohibitive cost, placing it out of reach for much of the academic and open-source communities. Although strong open-source efforts already exist, including open-weight models and open-source training recipes, a cost-efficient, hardware-accessible, and open-source pretraining recipe has long been missing. Even at a small scale, training Llama-3.2-3B costs over $1.5M, and reproducing SmolLM3-3B needs over $700K. In this report, we present an open pretraining recipe designed to lower this barrier. Using this recipe, we train a collection of Puro-2B models from scratch on up to 1.4 trillion tokens with FP8 precision on consumer-grade RTX 5090 GPUs. The models in the collection differ in token budgets and selected recipe variants. Our best model is trained at a compute cost of less than $6.9K and approaches Qwen2.5-1.5B performance under our evaluation protocol. This cost efficiency is enabled by a combination of approaches, including hardware selection, low-precision training, hyperball optimization, curriculum model averaging, and the data recipe. Beyond the recipe itself, we provide two additional results. First, across the Puro-2B collection, we derive a Puro Cost Scaling Law that relates training cost to average model performance; the fitted law suggests that about $4.4K, less than $5,090, is sufficient to reach the performance of Qwen2-1.5B. Second, as an end-to-end case study, we examine how pretraining data curricula shape downstream performance after post-training. Such controlled studies are enabled by having access to the full pretraining pipeline rather than model weights alone. We release the full training recipe for Puro-2B, including data, code, and model weights under Apache 2.0 at https://huggingface.co/collections/thu-pacman/puro-2b.

Kairong Luo, Jiarui Cui, Yaorui Yin +8

Tsinghua University · Pengcheng Laboratory

Language Model Pretraining Language Model Scaling Laws Efficient Language Model Training

Aug 12, 2026cs.LG

Small-Scale Experiments: Are We There Yet?

Scaling laws promised cost-effective experiments; six years later, they have yet to fully deliver. Instead, researchers have found them unreliable at small scales (starting at 4M parameters) and concluded that sizable models cannot be avoided. We show this is not the case: the confounding factor is hyperparameters. Small models are highly sensitive, but hyperparameter sensitivity fades with scale. This small-scale sensitivity makes scaling laws easy to miss because they only emerge on the fully tuned frontier, and reaching that frontier requires an extensive search far beyond what most ever run. By ablating the basic scaling law recipe, we show well-tuned hyperparameters matter more than any other ingredient. Further, we reveal why those hyperparameters become easier to find: as scale increases, the hyperparameter loss surface becomes lower dimensional. Nevertheless while scaling laws exist in small models, extrapolation hits statistical limitations. A holistic approach is required. Synthesizing our insights with the recent literature, we develop a new methodology for model-centric research and demonstrate it on a question that once took the field years to settle: where to place normalization layers in the transformer architecture. From small-scale experiments, we recover the large scale result: pre-normalization works better as models grow in size. With the right tools and a better understanding, small-scale experiments can deliver on scaling laws' long-awaited promise.

Nicholas Lourie, Kyunghyun Cho, Karen Ullrich +1

1FAIR at MSL Meta · 2New York University

Language Model Scaling Laws Hyperparameter Optimization

Aug 11, 2026cs.LG

Compute-Optimal Is Not Cluster-Optimal: Systems-Aware Scaling for Sparse Mixture-of-Experts

In large-scale pretraining, the algorithm, architecture, and systems decisions are conventionally made in disconnected stages. A scaling law stage selects an architecture and training recipe, optimizing loss under compute constraints, and a separate systems stage then optimizes the implementation for hardware efficiency. In this work, we develop MOSAIC, which formulates model architecture and systems co-design as an optimization problem. MOSAIC couples a predictive scaling law with a calibrated performance model that estimates Model FLOPs Utilization (MFU), communication cost, memory footprint, and the best parallel layout. We instantiate the framework for sparse Mixture-of-Experts (MoE) language models, where expert count, routing sparsity, and other MoE layer dimensions affect both the loss and systems efficiency. We fit a scaling law on sparse MoE models trained on text data, whose scaling dimensions include the sparsity factor, which is the fraction of model parameters inactive per token in a forward pass. The scaling law sweeps in our work span active parameters from

104

million to

2.7

billion and total model sizes reaching

79

billion parameters. We show that, within the calibrated sparsity range, an efficiency-agnostic model-FLOPs budget admits no interior optimal sparsity. The fitted loss decreases monotonically with sparser models and the compute optimum lies at the upper boundary of the data support. An optimal sparsity in MoE models instead emerges under the cluster's systems constraints, as captured by MOSAIC. Our results argue for a shift towards unified architecture and systems co-design for frontier language model training.

Soumajyoti Sarkar, Yuxin Tang, Sheng Zha

Amazon AGI Foundations

Mixture-of-Experts Language Models Language Model Scaling Laws Efficient Language Model Training Hardware-Software Co-Design

Aug 7, 2026cs.CL

Skaling: Chinchilla's Exponents Meet Kaplan's Coupling

Neural scaling laws are foundational for language model development, yet standard formulations systematically under- and overestimate loss at data-scarce and overtraining extremes. This failure originates in the underlying assumption that model size and training data impact the loss independently. To address this, we introduce the Skaling law, a generalized functional form that couples model capacity and data through a single interaction exponent. This simple extension reduces the Mean Absolute Percentage Error (MAPE) by 1.5-3x across both interpolation and extrapolation regimes. When paired with a sparse grid strategy restricted to low-compute regimes, the Skaling law achieves accurate full-grid extrapolation using approximately 10x less compute than uniform sweeps. By enabling reliable performance prediction from small-scale experiments, the Skaling law provides a more robust and resource-efficient framework for allocating compute budgets in next-generation model training.

Mathurin Videau, Badr Youbi-Idrissi, David Lopez-Paz +1

1FAIR at Meta

Language Model Scaling Laws Neural Scaling Laws

Aug 5, 2026cs.AI

Small Foundation Models of Human Cognition and Behaviour

Large language models fine-tuned on human behavioural data have emerged as general-purpose cognitive proxies, but the scale this requires, and whether these models process task structure or exploit statistical shortcuts, remain open questions. We train fourteen models from 135M to 14B parameters across four architecture families on Psych-101, a dataset of 10.7 million trial-level choices from 160 experiments. For in-distribution simulations, scale barely matters. The models fall within a narrow band, as though against a ceiling, and 0.6B to 1B parameters suffice to match a 70B baseline on held-out participants. Out-of-distribution, that band opens into a markedly steeper scaling gradient, with larger models clearly advantaged in generalisation to novel task structure. To determine what information these models use, we run two diagnostics. We progressively strip four prompt channels -- task instructions, experimental stimuli, outcome feedback, and choice history -- across 27 experiments, and permute trial order. Masking the content of stimuli and feedback destroys 75.7% of learned information and pushes models below chance, demonstrating that choice history alone does not account for performance. Permutation reveals invariance on tasks with independent trials but sensitivity where trial order is determined by prior responses. Small cognitively fine-tuned models therefore show promise as noise ceiling estimators for psychological experiments, though their scope remains bounded by the paradigms seen in training.

Nick Oh, Fernand Gobet

socius labs · London School of Economics and Political Science

Computational Cognitive Modeling LLM Fine-Tuning Language Model Scaling Laws

Aug 4, 2026cs.AI

LLaDA MoE v2: Scaling Mixture-of-Experts Diffusion Language Models

Diffusion language models (dLLMs) offer an alternative to autoregressive (AR) language modeling, yet the scaling behavior of Mixture-of-Experts (MoE) dLLMs remains poorly understood. We systematically characterize how optimization hyperparameters, compute allocation, and architecture scale for MoE dLLMs, identifying quantitative differences from scaling trends previously reported for AR models. Specifically, for optimization, the optimal nominal batch size grows faster, while the optimal learning rate decays more rapidly with compute. For model--data allocation, IsoFLOP analysis reveals a slight data-side tilt: the optimal token budget grows faster than activated model-side computation. For MoE architecture, larger scales increasingly favor larger expert pools at fixed activated capacity, while moderate expert granularity remains consistently effective and the preferred fraction of activated capacity assigned to shared experts remains stable across scales. Guided by these findings, we train LLaDA MoE v2, a 30B-A3B dLLM, from scratch on 23.5T tokens. With approximately 65% as many pretraining tokens as Qwen3, LLaDA MoE v2 approaches Qwen3 on several knowledge, reasoning, and coding benchmarks. After supervised fine-tuning alone, it outperforms SDAR Chat on seven of eight reasoning and coding benchmarks and remains close to Qwen3 on several tasks. These results establish practical scaling laws and design principles for MoE dLLMs.

Fengqi Zhu, Shaoxuan Xu, Jingyang Ou +11

Gaoling School of AI, Renmin University of China · Beijing Key Laboratory of Research on Large Models and Intelligent Governance · Engineering Research Center of Next-Generation Intelligent Search and Recommendation, MOE +2

Mixture-of-Experts Language Models Language Model Scaling Laws Diffusion Language Models

Jul 30, 2026cs.LG

Towards joint scaling laws with optimal batch size schedules

Modern deep learning typically keeps the batch size static throughout training, thus overlooking the joint effect of learning rate and batch size on the training dynamics. In this paper, we study the deep learning dynamics through the lens of convex optimization and derive a joint characterization of loss in terms of both schedules, applicable to general optimizers and model architectures. This characterization yields a closed-form optimal batch size schedule for any prescribed learning rate schedule, and further leads to joint scaling laws that consistently outperform static batch size baselines, highlighting the significance of dynamic batch size schedule in large language model training.

Jiaxiang Li, Zhiqi Bu, Shiyun Xu

Meta · Independent researcher

Deep Learning Optimization Language Model Scaling Laws Neural Network Training Dynamics LLM Training

Jul 28, 2026cs.LG

Bridging Compute- and Data-Optimal Pretraining

Classical compute-optimal scaling laws assume an unbounded supply of fresh pretraining data, yet pretraining is increasingly entering a regime in which compute grows faster than the availability of high-quality data. We propose Compute-Data (CD) scaling laws, a unified framework that bridges compute-optimal scaling, where data scales freely with compute, and data-optimal scaling, where the corpus is fixed while compute can grow without bound. CD scaling extends classical scaling laws by introducing a token-effectiveness function,

η

, which quantifies the value of a derived token-produced, for example, through multi-epoch repetition or paraphrasing-relative to a fresh token, ranging from a perfect substitute to having no value. We fit

η

for two data-expansion strategies, multi-epoch repetition and paraphrasing, across model sizes from 14M to 600M parameters using the Dolma-3 corpus. We find that token effectiveness is far from constant: it depends jointly on model size, the tokens-per-parameter ratio, and the amount of derived data, and it saturates as the corpus is expanded. The functional form of

η

implies diminishing returns when substituting compute for data as either model size or data availability increases. It also partitions training into three operational regimes---compute-bound, data-bound, and model-bound---and shows that classical compute-optimal allocation is suboptimal across most practically relevant settings.

Tian Qin, Kimia Hamidieh, David Alvarez-Melis

Harvard University · MIT CSAIL

Language Model Pretraining Language Model Scaling Laws Scaling Laws Data-Efficient Learning

Jul 18, 2026cs.AI

Lomekwi: Resource-Bounded Tool Discovery in LLM Agents

Existing tool-use benchmarks report a single success rate for complex, multistep tasks. Inspired by ideas from cognitive science, we distinguish tool use from tool discovery and decompose the latter into curiosity (the model's ability to discover the parts needed to build the tool), recognition (the model's ability to discover the process of creating the tool), and efficiency (the model's use of the tool after creation). We show that this framework can be applied to existing discovery tasks, such as Voyager. In addition, we provide evidence that recognition inversely scales with model size, and we introduce and analyze a class of combinatorial games that demonstrates this. We further observe inverse scaling in a separate environment designed to emulate real-world tasks.

Roshan Klein-Seetharaman, Daniel Wang, Andrew Xu

Sea12 Technologies · Yale University

Language Model Scaling Laws LLM Agent Evaluation AI Agent Benchmarks LLM Agents

Jul 13, 2026cs.LG

Domain-Aware Scaling Laws Uncover Data Synergy

Machine learning progress is often attributed to scaling model size and dataset volume, yet the composition of data can be just as consequential. Empirical findings repeatedly show that combining datasets from different domains yields nontrivial interactions. For instance, adding code improves mathematical reasoning, while certain mixtures introduce interference that reduces model performance. We refer to these effects collectively as data synergy, where the contribution of multiple domains exceeds or falls short of the sum of their isolated contributions. In this work, we formalize and quantify data synergy in language model pretraining. Leveraging observational variation across open-weight LLMs with diverse pretraining mixtures, we estimate both direct domain-to-benchmark synergy (how one domain contributes to performance on another) and a second-order domain-domain synergy (capabilities that require co-occurrence of multiple domains). Our framework improves predictive accuracy over domain-agnostic scaling laws and recovers stable synergy estimates. We validate these estimates by training models on predicted optimal and predicted anti-optimal mixtures and confirm that our synergy estimates correctly predict performance rankings.

Kimia Hamidieh, Lester Mackey, David Alvarez-Melis

MIT CSAIL · Microsoft Research · Harvard University

Language Model Pretraining Language Model Scaling Laws Data Mixture Optimization

Jul 9, 2026cs.CL

Hidden Decoding at Scale: Latent Computation Scaling for Large Language Models

Scaling Large Language Models (LLMs) has been driven mainly by enlarging the Transformer backbone, but for an already-strong model this requires another round of costly pretraining. We study whether an existing backbone can keep improving by allocating more computation to each token while leaving the Transformer backbone fixed. Depth-recurrent (looped) Transformers pursue this goal but are hard to scale, because looped computation does not fit naturally with the pipeline parallelism used to train the largest models. We add computation along the sequence-length dimension, where the extra computation is simply a longer input and stays compatible with standard large-model training. We propose Hidden Decoding, a sequence-length scaling method applied during continued pretraining (CPT). It expands each token into n streams with independent embedding tables and keeps the intermediate streams' key-value cache as context, so each token performs more internal computation without adding or widening Transformer layers. To keep this affordable at scale, we introduce Stream-Factorized Attention, in which most layers attend only within each stream and only a few layers mix across streams, reducing the attention cost from quadratic to roughly linear in n. Experiments support two scaling results. At frontier scale, we train WeLM-HD4-80B and WeLM-HD4-617B at n=4 and improve their matched non-HD baselines, making Hidden Decoding the first demonstrated sequence-length scaling method at the 100B+ MoE scale. Across expansion factors, the gains grow as n increases, showing that sequence-length expansion is a practical fixed-backbone scaling path for frontier-scale LLMs.

Aiwei Liu, Cheng Shi, Chuhan Wu +45

Language Model Scaling Laws Transformer Attention Language Model Scaling LLM Training

Jul 2, 2026cs.CL

Will Scaling Improve Social Simulation with LLMs?

Large Language Model (LLM) social simulations are a promising research method, but they are not yet faithful enough to be adopted widely. In this work, we investigate whether the current scaling paradigm in language modeling is likely to close these gaps, or whether simulation fidelity is orthogonal to general capabilities and therefore deserving of more research attention. We use scaling laws to study the relationship between LLMs' compute scale, general capability benchmarks, and the fidelity of social simulation in three representative sub-domains: opinion modeling, behavioral simulation, and longitudinal forecasting. Surprisingly, we discover strong compute scaling in all three settings, using a suite of 85 transformer LLMs with the Qwen3 architecture pre-trained on the DCLM web text corpus under fixed-compute budgets from

10^{18}

to

10^{20}

FLOPs. Then we evaluate 35 larger and more capable open-weight models up to 70B parameters, allowing us to predict downstream accuracy from loss. This reveals that the majority of behavioral and opinion simulation tasks will rapidly improve with scale, particularly when they involve populations that are well-represented in English web corpora. Longitudinal forecasting and underrepresented opinions scale more slowly, especially when they are less correlated with general knowledge and reasoning benchmarks like MMLU. In behavior simulation, scaling fails to improve model calibration with human cognitive biases like risk aversion, as well as human heuristics like learning correlated rewards from related tasks. On these tasks, even fine-tuned models fail to noticeably scale up performance from 0.5B to 8B parameters. Taken together, we conclude that scale will improve social simulations in most settings, but outliers exist, and improvements will be less reliable in low-resource domains.

Caleb Ziems, William Held, Su Doga Karaca +3

Stanford University · Open Athena

Opinion Dynamics Language Model Scaling Laws Computational Social Science Language Model Calibration Large Language Model-Based Social Simulation

Jul 2, 2026cs.CL

When Does Generating More Help? Disentangling Fixed-Source Synthesis from Source Expansion in Synthetic Data Scaling

Synthetic data can be scaled along two routes: Source Expansion (SE), which enlarges the source by adding seed materials or generators, and Fixed-Source Synthesis (FSS), which holds the source fixed and scales the generation budget. Existing scaling studies typically expand the source as the data grows, conflating SE with FSS and leaving FSS underexplored. We isolate FSS by holding the seed-question pool and teacher model fixed, varying only the per-question response budget under Rejection Sampling (RS). We adapt the rectified scaling law to FSS, deriving it from how repeated sampling covers a fixed source. Empirically, the derived form, fit on low budgets, predicts performance at the held-out highest budget for every evaluated teacher--student pair. At matched total-sample budgets, SE and FSS are comparable at small budgets; at large budgets, adding seed questions outperforms spending the same budget on more responses. Within FSS, however, neither synthesizing additional questions from the existing seeds nor varying the synthesis protocol outperforms plain RS at matched budgets. FSS is thus a bounded scaling axis and a controlled setting for comparing synthesis protocols. We will release our code and data to facilitate further research.

Xu Guo, Jian Tong, Zhihui Lu +1

Shanghai AI Laboratory · Shanghai Innovation Institute · Fudan University

Synthetic Data Generation Language Model Scaling Laws Synthetic Data Generation for Language Models Rejection Sampling

Jul 1, 2026cs.LG

How to Allocate Your Tokens? Scaling Laws with Training Steps and Batch Size

We propose a scaling law that takes into account model size and training data while explicitly splitting the latter into training steps and batch size (called three-term law). Fitting the proposed law on a large set of training runs, we find that it correctly recovers the scaling of the optimal batch size. Moreover, because it makes use of training runs with suboptimal batch size, our proposed law can be robustly fit with a significantly smaller amount of training runs. We further show that the three-term law can be used to derive scaling laws for suboptimal batch sizes, and that it matches previous empirical findings related to the critical batch size.

Fabian Schaipp

Inria, ´Ecole Normale Sup´erieure, PSL Research University, Paris

Language Model Scaling Laws Scaling Laws Neural Scaling Laws Hyperparameter Optimization

Jul 1, 2026cs.AI

Two AI Metrics Diverged: Will it Make All the Difference?

As exponential compute scaling continues, will the capabilities of frontier AI models outstrip what is accessible to developers on a small fixed budget? Or will capabilities converge, with "meek models inheriting the earth"? Building on Gundlach et al. (2025b), we show that the answer depends on how we value and measure AI capabilities. We discuss conventional performance measures and show that, while validation loss shows a shrinking gap, on other metrics frontier models grow their lead forever. Classifying performance metrics by their functional forms in relation to training (and inference) compute, we provide tight mathematical conditions for determining which metrics favor meek models, and show that bounded performance metrics always do. But careful interpretation of performance metrics is essential: we show that many common bounded metrics have closely-related counterpart metrics that are unbounded (and vice versa). Determining the apt metric in a domain is a prerequisite for policy, since bounded and unbounded metrics may suggest opposing policy responses. If a particular capability -- like software engineering, synthetic biology, or rhetorical persuasiveness -- is unbounded when measured in the terms we care about, frontier-level capability will likely be concentrated in the hands of a few wealthy actors. Conversely, if that capability is instead bounded, frontier-level capabilities proliferate through meek models into the hands of the many.

Alex Fogelson, Zachary A. Brown, Hans Gundlach +2

MIT FutureTech, CSAIL, Cambridge, MA, USA.

Language Model Scaling Laws AI Governance AI Scaling Laws

Jun 29, 2026cs.CL

Smooth Scaling Laws Hide Stepwise Token Learning

Language model loss follows remarkably regular scaling laws over model and data size, yet it remains unclear why the aggregate loss should exhibit a power-law form. Existing explanations often attribute this regularity to a heavy-tailed spectrum of pattern difficulty in natural language, but this view has not been directly validated at token-level granularity in large-scale real-data training. We present a token-level framework that decomposes scaling laws into localized learning events of individual contextualized tokens. By fitting token loss trajectories with sigmoids, we show that token learning is concentrated in localized transitions, giving rise to a learning-time spectrum that dominates the scaling-law shape. Across more than one hundred pre-training runs on large and diverse real-language corpora with modern LLM architectures, scaling up to 6B parameters and 300B training tokens, the measured learning-time spectrum quantitatively reconstructs the validation loss derivative along the training-step

T

, data-scale

D

, and model-scale

M

axes. We further show that the same signal is actionable: by reshaping the training distribution according to when tokens become learnable, we alter the optimization trajectory and achieve 11% faster validation-loss reduction. These results provide direct empirical evidence that scaling laws are governed primarily by the distribution of token-level learning times, and that this distribution can be used not only to explain scaling behavior but also to improve training performance.

Pingjie Wang, Zechen Hu, Peiru Yang +2

Dots Studio, Xiaohongshu Inc. · Shanghai Jiao Tong University · Tsinghua university

Language Model Pretraining Language Model Scaling Laws LLM Training

Jun 28, 2026cs.LG

Representational Depth of Evaluation Awareness Shifts With Scale in Open-Weight Language Models

Do language models know when they are being tested? This question matters for AI safety: a model that recognises an evaluation context could alter its behaviour strategically, making downstream benchmarks harder to interpret. Using 11 models spanning Qwen 2.5, Gemma 2, and Llama 3.2, we find a systematic size-dependent shift in representational depth: in both Qwen 2.5 and Gemma 2, the layer at which evaluation-awareness is most linearly recoverable moves from late layers in smaller models to early layers in larger ones. This suggests that scale changes not only the strength of evaluation-awareness but also where it is most linearly recoverable in the network. This depth shift helps explain why within-family scaling trajectories are non-monotonic or inverse rather than smooth and family-general, showing that a simple universal power-law account is not supported under denser within-family sampling. Finally, white-box probe signals are consistently stronger than black-box behavioural expression, and the relationship between the two varies by family in ways not predicted by probe AUROC alone.

Archit Manek

Independent Researcher.

LLM Interpretability Language Model Scaling Laws Evaluation Awareness in Language Models Metacognition in Language Models Language Model Probing

Jun 28, 2026cs.LG

On the Nonlinearity of Learning Rate Scaling for LLM Training

Learning-rate transfer can reduce the cost of training large language models: instead of sweeping learning rates at target scale, practitioners extrapolate from smaller runs. Existing approaches often assume that the optimal learning rate follows a log-linear scaling law in data scale and model size. We carefully examine and evaluate this scaling law. In our empirical study of GPT-2--style models from 22M to 707M parameters trained on 5B to 100B tokens, the optimal learning rate develops upward curvature at larger scales, leading to inaccurate extrapolation. We find that this curvature largely disappears when learning rates are replaced by effective learning rate (the step size in normalized weight space), and when data

D

extrapolation is used instead of model size

N

extrapolation. Next, we explain nonlinearity in scaling: weight-norm converges to equilibrium slower when optimal learning is small, requiring a larger step size to reduce the transient phase. Experiments with AdamH, which directly controls the effective learning rate, further support this explanation.

Zaiwen Yang, Huaqing Zhang, Jing Xu +1

Tsinghua University · Tsinghua University, Shanghai Qi Zhi Institute

Language Model Scaling Laws Hyperparameter Transfer Neural Network Training Dynamics LLM Training

Jun 26, 2026cond-mat.dis-nn

Scaling limit of the Random Language Model

We develop a quantitative theory of the Random Language Model (RLM), an ensemble of stochastic context-free grammars, in a scaling limit where the number of hidden symbols

N \to \infty

while the grammar temperature

\tildeε_d \to 0

at fixed

x = {\tildeε}_d \log N

. In this limit, the model admits a controlled description based on a large-deviation principle over rule-usage patterns. A semi-annealed approximation maps the problem to a class of Random Energy Models with nontrivial combinatorics. We show that the RLM exhibits a condensation transition at a critical value

x_c=1/8

, below which rule usage concentrates and language statistics acquire a nontrivial dependence on corpus length. A second characteristic scale at

x=1/2

marks the onset of entropy reduction from its maximal value. Across these regimes, we derive explicit scaling laws for the number of distinct rules, entropy, and related observables, identifying distinct scaling, saturation, and critical regimes controlled by the interplay of grammar size, corpus length, and temperature. The theory resolves previous ambiguities regarding the existence of a thermodynamic transition and explains the slow approach to the large-

N

limit as a consequence of the dependence on

\log N

. It further provides a unified framework in which universal statistical properties of language emerge from typical realizations of generative grammars, with implications for both natural language statistics and the behavior of large language models.

Eric De Giuli

Department of Physics, Toronto Metropolitan University, M5B 2K3, Toronto, Canada

Language Model Scaling Laws Language Modeling

Jun 23, 2026cs.LG

Emergent Capabilities Arise Randomly from Learning Sparse Attention Patterns

Neural scaling laws for transformer language models predict smooth improvements in pretraining loss with increasing parameters, but downstream capabilities such as in-context learning are known to emerge abruptly past a certain model scale. In this paper, we show that emergent capabilities arise stochastically throughout training, with larger models acquiring them earlier on average. We demonstrate that the emergence of capabilities such as pattern completion and indirect object identification corresponds to the abrupt learning of task-relevant attention patterns. To isolate this phenomenon, we train transformer models on synthetic linear map and cellular automata datasets, and we show that the difficulty of learning attention patterns depends on context length and pattern sparsity. Moreover, scaling the number of attention heads improves learning efficiency on our synthetic tasks, while increasing the head dimension yields diminishing returns past a minimum capacity. We additionally investigate architectures with alternative attention mechanisms, showing that MLP-Mixer outperforms a transformer on linear map tasks with complex attention patterns. Our findings provide a mechanistic insight into emergence, showing that downstream capabilities arise abruptly due to the intrinsic difficulty of learning sparse attention patterns in transformer models.

Vatsal Baherwani, Zixi Chen, Shikai Qiu +2

New York University

Language Model Scaling Laws Transformer Attention Sparse Attention

Jun 23, 2026cs.LG

Neural Scaling Universality: If Exponents Are Fixed, Time to Understand Coefficients

Neural scaling laws describe how pre-training loss decays as power laws with training time, model size, and compute. This position paper argues that the exponents of these power laws are fixed by generic mechanisms: a one-third time scaling due to the strong nonlinearity of Softmax, an inverse width scaling due to representational superposition, and an inverse depth scaling due to ensemble averaging of Transformer layers. These mechanisms are robust to a wide range of data structures and architectural details, placing current large language models in a universality class with fixed exponents. The coefficients, however, are expected to be sensitive to data and architecture details, and directly determine practical quantities such as the optimal model shape and the compute-optimal frontier. We therefore argue that understanding the coefficients is the key to near-term performance improvements, and that a closer examination of the current universality class may reveal pathways to better universality classes.

Yizhou Liu, Jeff Gore

Massachusetts Institute of Technology Cambridge, MA 02139

Language Model Scaling Laws Neural Scaling Laws

Jun 23, 2026cs.AI

Can Scale Save Us From Plasticity Loss in Large Language Models?

The loss of plasticity - the ability of a network to learn new information after having already learned older information - is a fundamental challenge in creating artificial neural networks capable of continual learning. Although this phenomenon has been known for decades, it has mostly been studied in older, relatively small architectures and rarely in natural-language domains. To determine whether loss of plasticity remains a problem in the modern transformer-based LLM paradigm, we study plasticity loss in GPT-style Transformer models trained on a multilingual continual learning problem. Consistent with prior work, we find evidence of plasticity loss across models ranging from 5M to 314M non-embedding parameters, as measured by deterioration on a held-out Vietnamese probing task. We further find that the onset of plasticity loss follows a predictable scaling law, growing sublinearly with model size. These results suggest that larger models may delay the measurable effects of plasticity loss, but that increasing parameter count alone is likely to be insufficient to completely prevent it. We also find evidence of plasticity loss under stationary multilingual training, challenging the view that the phenomenon is exclusive to continual learning with abrupt task changes. Overall, our results suggest that even large Transformer language models trained on natural-language will eventually lose the ability to efficiently adapt to new data after sufficiently long training, in both continual and stationary settings.

J. Fernando Hernandez-Garcia, Tomás Figliolia, Beren Millidge

Zyphra San Francisco, CA

Continual Learning for LLMs Loss of Plasticity Language Model Scaling Laws

Jun 23, 2026cs.AI

Scaling Laws for Task-Specific LLM Distillation

Large Language Models (LLMs) achieve strong performance across a growing range of domains, yet their scale poses deployment challenges in applications where latency and cost constraints are critical. This paper derives empirical scaling laws for domain-specific LLM compression, quantifying how in-domain and general knowledge performance scale with dataset size, compression ratio, supervision format, and iterative pruning schedule. Using quantitative finance as our application domain, we compare logit-based and LoRA-based distillation under iterative structural pruning, introducing a blended chain-of-thought supervision loss that stabilizes KL-divergence distillation over reasoning traces. In-domain task quality degrades predictably under compression while general-knowledge benchmarks collapse well before the same point; supervision format is the key driver of this tradeoff, with chain-of-thought supervision actively recovering general knowledge that pruning erases. We release the headline dataset FinHeadlineMix, scaling law results, and practical recommendations to provide a reusable framework for domain-specific compression decisions.

Lavinia Ghita, Dhruv Desai, Ioana Boier

NVIDIA

LLM Pruning LLM Compression Language Model Scaling Laws Language Model Distillation

Jun 23, 2026cs.LG

Internal Data Repetition Destroys Language Models

Language models are running out of high-quality training data, and even aggressively deduplicated corpora retain some amount of repetition. Earlier controlled studies predated Chinchilla-style scaling laws and could only measure the cost of repetition indirectly. We revisit repetition in the Chinchilla era, using a fitted no-repetition scaling law to report Compute-Equivalent Gain and Compute-Equivalent Loss. We show that under this modernized paradigm, repetition damage is systematic in three ways. First, holding compute allocated to repeated data constant, eval loss peaks at an intermediate repeat count

\Rep

; repeating a moderately sized subset a moderate number of times damages performance more than repeating a large subset a few times or a small subset many times. Second, the location of this peak is well-fit by a power law in model size; this scaling law reveals that the most damaging number of repeated data grows more quickly than compute. Finally, when repeated documents consume 10% of the FLOPs budget in a controlled exact-document repetition setting, the compute-equivalent loss can be large: on FineWeb-Edu-Dedup, the most damaging repeat count for a Qwen3-style 344M-parameter model at

\OT=1

matches the loss of a no-repetition run using 67% of the FLOPs. We demonstrate that these phenomena are not language-model-specific, and can be analytically understood in a simple statistical model: a misspecified linear regression with verbatim duplicates reproduces the same qualitative loss peak, quantifying how such peaks can arise from a statistical tradeoff between memorization and generalization. Our findings add precision to the study of duplication in language models, allowing practitioners to quantify the wasted compute incurred by the presence and repeat structure of duplicates in pretraining corpora.

Jessica Chudnovsky, Joshua Kazdan, Noam Levi +6

1Stanford Computer Science. · 2Stanford Statistics. · 3Tel Aviv University. +1

Memorization in Language Models Language Model Scaling Laws LLM Training Data Repetition in Language Model Training

Jun 23, 2026cs.AI

On the Smallness of the Large Language Models Scaling Exponents

We discuss reasons why the scaling exponents of current Large Language Models (LLMs) applications are indicating an unsustainable regime in terms of energy resources. We further show that attributing the smallness of such exponents to a numerical bias due to the neglect of a non-zero value of the loss function in the limit of infinite data (``pedestal effect") does not remove the unsustainability issue. Finally, the effects of the smoothness (roughness) of the data on the scaling exponents is commented upon based on an analogy with phenomenological models of fluid turbulence.

Sauro Succi, Peter V. Coveney, Alex Hansen

Italian Institute of Technology, Viale Regina Elena, 291, 00161, Rome, Italy · PoreLab, Physics Department, Norwegian University of Science and Technology, 7491 Trondheim, Norway · Centre for Computational Science, Chemistry Department, University College of London, 20 Gordon Street, WC1H 0AJ, London, United Kingdom

Energy-Efficient ML Language Model Scaling Laws

Jun 22, 2026cs.LG

The Energy Consumption of Transformer Fine-Tuning: A Roofline-Inspired Scaling Model

Transformer-based models underpin modern natural language processing but incur rapidly growing computational and energy costs. As training scales in both model size and parallelism, accurately predicting energy consumption has become critical for sustainable and cost-aware system design. We present a framework for modeling the energy consumption of Transformer training on multiple GPUs. Using controlled architectural sweeps of BERT models, we relate measured energy to lightweight proxies for compute, memory traffic, and hardware efficiency. Inspired by roofline models, our approach incorporates a speedup-based hardware-efficiency factor that captures the effects of tensor parallelism and fully sharded data parallelism. We derive a scaling law model that accurately predicts training energy across heterogeneous configurations.

Mansour Zoubeirou a Mayaki

Université Lumière Lyon 2, CNRS, Ecole Centrale de Lyon, INSA Lyon, Universite Claude Bernard Lyon 1, LIRIS, UMR5205, 69007 Lyon, France

Energy-Efficient ML Transformer Language Model Scaling Laws

Jun 20, 2026cs.AI

Human vs Machine Mathematical Difficulty on Project Euler: An Experimental Analysis

We study how the effort and success probability of frontier AI systems scale with human difficulty on problems from Project Euler, an online platform of computational mathematics problems. Our dataset, from the MathArena benchmark, consists of 3840 attempts across 50 problems and 26 model configurations, with problem difficulty measured by the site's public human solve times. Motivated by a proposal of Timothy Gowers, we test a power-law relation

t_{\text{machine}} = a \cdot t_{\text{human}}^b

between generated-token cost per successful answer and human time, and find

b < 1

for 20 of the 25 models with usable fits, including the strongest base models; this operationalization therefore does not support an earlier prediction that machines scale worse than humans with difficulty. We also investigate whether success probability on the tested problems can be modeled by a simple exponential decay

p_{\text{success}} = e^{c t_{\text{human}}}

, predicting a linear relation between

\log p_{\text{success}}

and

t_{\text{human}}

. Using a binning approach for data aggregation we find moderate empirical support (median bin-level

R^2 = 0.92

across the 22 best-covered configurations) for this model. Following METR, we also fit logistic success curves and extract 50% task-length horizons

h_{50}

; the strongest configurations in our 20 April 2026 snapshot reach roughly

2.5

--

4.3

hours on our fastest-five human baseline, with a log-linear fit through the state-of-the-art frontier giving a descriptive doubling time of about

75

~days for the SOTA

h_{50}

.

David Holmes, Johannes Schmitt

Mathematical Reasoning Benchmarks Language Model Scaling Laws LLM Mathematical Reasoning

Jun 5, 2026cs.LG

Data-Constrained Language Model Pretraining: Improved Regularization and Scaling Laws

Classical scaling laws for language model pretraining balance model size against training dataset size under a fixed compute budget, assuming abundant data and a single pass over the corpus. As training compute grows faster than the supply of natural language data, pretraining is likely to enter a data-constrained, compute-rich regime where models train for multiple epochs over a finite dataset. We study data-constrained pretraining along two axes, regularization and scaling. For regularization, we study masked-input regularization (MIR), an auxiliary next-token prediction loss on randomly masked inputs. MIR tests whether the random masking central to diffusion language models can benefit autoregressive pretraining without architectural changes or inference overhead. Across 72M to 1.4B parameter models, we find that MIR added on top of strong weight decay improves validation loss over autoregressive strong-weight-decay-only models, with downstream gains at 1.4B. For scaling, we propose SoftQ, a scaling law that couples model size and data size to capture their interaction under repeated data. Classical alternatives such as the Chinchilla law use an additive form that decouples these terms, making them misspecified in the data-constrained regime. We find that SoftQ fits data-constrained experiments substantially better than these alternatives, and estimates MIR's gains as equivalent to roughly 1.3 times as much unique training data. We release our code at https://github.com/yixinw-lab/dc_pretrain.

Zhiwei Xu, Shihao Wu, Hanseul Cho +2

University of Michigan · KAIST AI

Language Model Pretraining Language Model Scaling Laws Scaling Laws Data Repetition in Language Model Training

Jun 4, 2026cs.CL

Decomposing Factual Sycophancy in Language Models: How Size and Instruction Tuning Shape Robustness

Factual sycophancy occurs when a language model abandons a correct, verifiable answer under social pressure. Because a flip occurs only when pressure toward a false answer exceeds the model's neutral preference for the truth, flip rates conflate two mechanisms: the strength of that baseline preference (truth margin), and how far pressure shifts it (manipulation sensitivity). We decompose factual sycophancy into these channels and use them to separate the effects of size and instruction tuning across 56 open-weight models spanning 0.3B-32B parameters and 13 manipulation types. We find that vulnerability is governed mainly by size, but instruction tuning changes how size acts: small instruction-tuned models can become less robust, whereas large instruction-tuned models usually become more robust. Instruction tuning primarily increases truth margin, but its behavioral effect depends on manipulation type. Scaling also changes the two channels differently: base models gain margin but become mildly more manipulation-sensitive, whereas instruction-tuned models gain margin faster and become less sensitive. Factual sycophancy is therefore not a single scalar property. Evaluations should report channel-specific, manipulation-specific, and size-conditioned robustness rather than flip rates alone.

Victor De Marez, Luna De Bruyne, Walter Daelemans

Centre for Computational Linguistics, Psycholinguistics and Sociolinguistics University of Antwerp Antwerp, Belgium

LLM Evaluation LLM Sycophancy Language Model Scaling Laws LLM Reliability Instruction Tuning Language Model Robustness

Jun 4, 2026cs.CL

Entity tracking emerges in sub-billion parameter language models and exceeds human performance in naturalistic narratives

Understanding language requires tracking entities across discourse - i.e., knowing where things are and how they change, even when not explicitly stated. Whether language models perform such tracking in a human-like fashion remains unclear, in part because existing evaluations rely on artificial tasks, far removed from natural language comprehension, and lack comparisons to humans. Here, we evaluate entity tracking in both language models and humans (N = 48) using naturalistic narratives at multiple levels of complexity. In humans, we find that entity tracking degrades specifically with narrative complexity, not narrative length. In language models, we find that human-level entity tracking is already present at 410 million parameters - well below the multi-billion parameter, code-specialised models identified by prior work - and improves with scale, with contemporary models far exceeding human performance. Together, these results demonstrate that entity tracking, a core component of language understanding, emerges at model scales far smaller than previously thought.

Karolina Drożdż, Micha Heilbron

IDEAS Research Institute; Warsaw, Poland · Max Planck Institute for Psycholinguistics; Nijmegen, Netherlands · University of Amsterdam, Amsterdam Brain and Cognition; Amsterdam, Netherlands

LLM Evaluation Language Model Scaling Laws

Jun 4, 2026cs.CL

Predictable Scaling Laws of Optimal Hyperparameters for LLM Continued Pre-training

The efficacy of continued pre-training for Large Language Models (LLMs) hinges upon hyperparameter configurations, such as learning rate and batch size. However, current practices often rely on heuristics or grid searches, leading to training instability and excessive costs. In this work, we first empirically discover that optimal hyperparameters follow stable and predictable scaling laws throughout the continued pre-training process. Leveraging these insights, we propose a novel framework to establish quantitative relationships between compute budget and optimal hyperparameters for a given checkpoint. Our approach has two stages: (1) \textit{Empirical Law Discovery}, where we train small-scale proxy models to derive functions mapping compute budget to optimal hyperparameters via standard loss-compute scaling laws; and (2) \textit{State-Aware Hyperparameter Prediction}, where we evaluate an initial checkpoint's validation loss and use the inverse scaling law to estimate its \textit{equivalent pre-training compute} -- the compute needed to achieve the same loss from scratch. Combining this with the planned compute budget, we predict optimal hyperparameters for the target run. Empirical results demonstrate that our method reduces the hyperparameter search overhead by up to 90% while achieving comparable or superior performance relative to baselines. This model-agnostic framework generalizes across architectures, providing a principled and efficient methodology for diverse continued pre-training scenarios starting from any given point.

Yongwei Zhou, Juncheng Diao, Junlin Shang +2

MeiTuan · University of Chinese Academy of Sciences · Harbin Institute of Technology

Language Model Scaling Laws Hyperparameter Transfer Continued Pretraining Hyperparameter Optimization

May 31, 2026cs.LG

Structure and Scale in Simplicial Sequence Modelling

Modern large-scale deep learning exhibits two striking empirical phenomena: behavioural scaling laws (predictable performance gains with increasing scale) and emergent mechanisms (structured internal representations and circuits in deep neural networks). We hypothesise that these two phenomena are connected: that predictable changes in behaviour are the result of predictable changes in internal computational structure. In this paper, we report preliminary evidence of such a connection. We find a correlation between scaling patterns in performance and representations in small transformers trained to predict the outputs of a hidden Markov model, for which residual activations are known to linearly encode a belief distribution over latent states in a probability simplex.

Matthew Farrugia-Roberts

Department of Computer Science, University of Oxford

Representation Geometry Transformer Language Model Scaling Laws Neural Network Interpretability Neural Scaling Laws Sequence Modeling

May 31, 2026cs.LG

When Data Is Scarce: Scaling Sparse Language Models with Repeated Training

Scaling laws for dense LLMs under infinite data are well explored, but how sparsity interacts with limited data is not. In this work, we study sparse training in data-constrained regimes where limited unique tokens require multi-epoch training. Our experiments span models up to 1.92B parameters in the fitting set, sparsity up to 93.75%, unique data budgets up to 2.6B tokens, and total training tokens up to 41.6B over 16 epochs; we further validate extrapolation on held-out dense-equivalent models up to 7.68B parameters. We find that: 1. Sparse scaling in data-limited settings: We introduce a scaling law that models loss as a function of active parameters, unique tokens, data repetition, and sparsity, accurately predicting performance across compute and data budgets. 2. Delayed data saturation: sparse training postpones diminishing returns from repeated data, making multi-epoch training more effective. 3. Resource trade-offs: With fixed data, loss-optimal sparsity is moderate ~ 50%, while compute-optimal sparsity is higher and grows with data scale. Overall, sparsity is not just a tool for efficiency, but a mechanism for improving scaling trade-offs under data scarcity. Our code is available at: https://github.com/boqian333/sparse-dc-scaling.

Boqian Wu, Qiao Xiao, Patrik Okanovic +6

University of Luxembourg · University of Twente · Eindhoven University of Technology +1

Language Model Scaling Laws Efficient Language Model Training Data Repetition in Language Model Training Data-Efficient Learning

May 29, 2026cs.LG

Item Response Scaling Laws: A Measurement Theory Approach for Efficient and Generalizable Neural Scaling Estimation

Scaling laws provide a fundamental framework for understanding the performance of Language Models (LMs), yet deriving them requires prohibitively expensive evaluations across thousands of checkpoints or millions of inference samples. To address this, we introduce Item Response Scaling Laws (IRSL), a unified framework that integrates Item Response Theory (IRT) within the scaling law framework. Unlike traditional approaches that treat each model-benchmark pair in isolation, IRSL disentangles latent model ability from question characteristics, factorizing the scaling law estimation for

M

models and

N

questions to significantly reduce parameter complexity from

O(M \times N)

to

O(M + N)

. We instantiate IRSL with Beta-IRT, which leverages the empirical probability responses of LMs -- such as token probabilities in pre-training and pass rates in test-time sampling -- to capture richer signals than binary responses. We validate our approach across two prevalent scaling paradigms: (1) pre-training downstream scaling, using 6,612 LM checkpoints and 37,682 questions from 10 benchmarks; and (2) test-time scaling, using 12 LMs and 120 questions from 4 benchmarks with up to 2,500 samples per question. Given a one-time calibration on existing model responses, IRSL yields more reliable scaling estimates using only 50 questions per benchmark (a 99.9% reduction), achieving comparable or superior decision accuracy to traditional approaches. Furthermore, we show that the estimated latent model abilities are generalizable, enabling accurate performance forecasting across benchmarks that share the same measurement objective.

Sang Truong, Yuheng Tu, Rylan Schaeffer +1

1Stanford University · University of California, Los Angeles

Language Model Scaling Laws Test-Time Scaling Item Response Theory

May 28, 2026cs.CL

How LoRA Remembers? A Parametric Memory Law for LLM Finetuning

Large Language Models (LLMs) must continuously learn and update knowledge to remain effective in dynamic real-world environments. While Low-Rank Adaptation (LoRA) is widely used for such memory updates, existing studies mainly rely on qualitative downstream evaluations, leaving the quantitative capacity limits and underlying dynamics of exact parametric memory largely unexplored. To bridge this gap, we employ LoRA as a controlled memory capacity probe within the latent space to systematically quantify exact parametric memory. We introduce the Parametric Memory Law, a robust power law linking loss reduction Delta L to effective parameters and sequence length. At the token level, fine-grained analysis reveals a deterministic phase transition, demonstrating that a prediction probability of p > 0.5 constitutes a sufficient condition for verbatim recall under greedy decoding. Driven by these insights, we introduce MemFT, a threshold-guided optimization strategy that dynamically redistributes the training budget toward sub-threshold tokens. Empirical evaluations demonstrate that MemFT can enhance memory fidelity and efficiency. Code will be released at https://github.com/zjunlp/ParametricMemoryLaw.

Ziwen Xu, Haiwen Hong, Linsong Yu +4

Zhejiang University · Alibaba Group

Memorization in Language Models LLM Fine-Tuning Language Model Scaling Laws LLM Memory Memorization in Generative Models Parameter-Efficient Fine-Tuning

May 28, 2026cs.LG

Why Larger Models Learn More: Effects of Capacity, Interference, and Rare-Task Retention

Larger models learn tasks smaller models do not. What drives this phenomenon? We develop a simple phenomenological argument that power-law scaling already suggests that a larger model will be able to learn a part of the data distribution that a smaller model fails to learn, even with infinite training data. To validate this claim and identify its causes, we study the effects of model scaling on a synthetic setup consisting of a mixture of tasks that show monotonic scaling curves. The results point to a data-induced competition over resources (neurons). Specifically, smaller models allocate their neurons to high frequency or low complexity tasks, and so they learn solutions that perform poorly on rare and complex tasks. Moreover, this happens even when solutions capable of expressing the desired task exist. We then assess how a larger model circumvents this data-centric bottleneck, finding that it traces to a reduced interference mechanism: larger models can allocate enough resources to common tasks that the gradient updates for those tasks become weak, which means that they do not overwrite rare-task features as they slowly accumulate. Finally, to further validate these claims, we pretrain OLMo models (4M to 4B parameters) on novel tasks of varying frequency and complexity. The results mirror those from our synthetic data experiments: only the larger OLMo models learn the infrequent and complex tasks, and these larger models embed more task features in their representations and show less gradient interference between tasks. Overall, we offer a data-centric account of why larger models learn tasks that smaller models fail to. This helps explain why larger models are better in practice, and it can inform practical questions concerning model sizing and training data mixtures.

Jing Huang, Daniel Wurgaft, Rachit Bansal +6

Stanford University · Kempner Institute at Harvard University · MIT +1

Long-Tail Learning Language Model Scaling Laws LLM Training

May 28, 2026cs.AI

The Curse of Helpfulness: Inverse Scaling Law in Robustness to Distractor Instructions via DistractionIF

Large Language Models (LLMs) are increasingly deployed in agentic and retrieval-augmented generation (RAG) systems, where they must execute user-specified tasks over externally provided reference text. In practice, such context is often unstructured and contaminated with benign but instruction-like semantic noise, such as editorial comments and system traces, which should be treated strictly as data. We introduce DistractionIF, a benchmark designed to evaluate robustness against such distractor instructions in reference text. Across a broad range of models, we observe a consistent inverse scaling phenomenon: larger models are often less robust, with performance dropping by up to 30 points as scale increases. Mechanistically, our perplexity analysis reveals that scaling erodes the probabilistic boundary between robust and distracted behaviors, making models increasingly prone to over-interpreting noise as instructions. To address this, we demonstrate that reinforcement learning, specifically Group Relative Policy Optimization (GRPO), can restore this boundary, improving robustness by up to 15.5% without compromising general instruction-following capability. Our findings highlight a critical instruction-following robustness gap in reference-grounded tasks and establish reinforcement learning as a promising path for enforcing strict data-instruction separation at scale.

Zeli Su, Zhankai Xu, Tianlei Chen +4

Minzu University of China, Beijing, China · Ant Group, Hangzhou, China · Renmin University of China, Beijing, China +1

Language Model Scaling Laws Language Model Robustness

May 27, 2026cs.CL

SuperValid: Capability-Aligned OOD Validation for Generalizable Downstream Scaling

Scaling laws guide large language model training by relating compute to cross-entropy loss, and recent work further extends them to predict downstream benchmark performance. However, prior approaches face generalization limitations from two aspects: focusing on benchmark-level performance introduces scenario-specific artifacts, while relying on IID validation loss fails to track capability improvements when training distributions vary. In this work, we argue that downstream scaling should be studied at the capability level, which captures shared skill factors across related tasks while abstracting away benchmark-specific noise. We propose SuperValid, a framework that synthesizes OOD (out-of-distribution), capability-aligned validation data by distilling core concepts from benchmarks within a capability domain and expanding them into diverse, knowledge-rich texts. Extensive experiments spanning 16 benchmarks grouped into 6 capability domains show that SuperValid loss exhibits strong and stable correlation with downstream performance across models of different architectures, scales, and training data distributions. As a training-free metric computable during training without benchmark evaluation, SuperValid enables effective model selection, early stopping, and scaling decisions.

Quanen Sun, Changxin Tian, Ke Shi +6

Ant Group

LLM Evaluation Language Model Scaling Laws OOD Generalization

May 27, 2026cs.LG

Law of Neural Interaction: Depth-Width Shape, Interaction Efficiency, and Generalization

The guidance of scaling laws has increased the resource demands of modern large language models (LLMs), yet it remains questionable whether these models utilize resources effectively under a fixed budget. Previous research has proved superposition as a key contributor to loss. By leveraging the Neural Feature Ansatz, we extend superposition from parameter space to gradient space and define it as neural interaction. We find that under a fixed budget, good generalization is usually accompanied by efficient neural interactions, and the model can be placed in an efficient interaction interval by adjusting its depth-width ratio (

R_{D/W}

). In addition, as the budget scales up, the efficient interaction interval of the model remains relatively stable. By comparing existing small scale dense LLMs, we observe that models operating near this interval tend to perform better on the MMLU-Pro benchmark. Our findings reveal that the

R_{D/W}

influences resource utilization efficiency and thereby affects generalization, providing insights into model shape initialization and the understanding of model generalization mechanisms. Code for Neural Interaction Law is available at: https://anonymous.4open.science/r/Neural_Interaction_Law-D788

Wenjie Sun, Jinning Yang, Shuai Zhang +1

The Chinese University of Hong Kong, Shenzhen · Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences · New Jersey Institute of Technology

Neural Network Generalization Language Model Scaling Laws

May 25, 2026cs.LG

Unified Neural Scaling Laws

We present a functional form (that we refer to as a Unified Neural Scaling Law (UNSL)) that accurately models and extrapolates the scaling behaviors of deep neural networks as multiple dimensions all vary simultaneously (i.e. how the evaluation metric of interest varies as one simultaneously varies the number of model parameters, training dataset size, number of training steps, number of inference steps, amount of compute, and various hyperparameters) for various architectures and for each of various tasks within a varied set of upstream and downstream tasks. This set includes large-scale vision, language, math, and reinforcement learning. When compared to other functional forms for neural scaling, this functional form yields extrapolations of scaling behavior that are considerably more accurate on this set.

Ethan Caballero, Priyank Jaini, David Krueger +1

Mila, University of Montreal · Google DeepMind

Language Model Scaling Laws Scaling Laws Neural Scaling Laws